A nearsighted man can clearly see objects up to a distance of 1.5m. Calculate the power of the lens of the spectacles necessary for the remedy of this defect.

Options

A. `-0.37D.`
B. `-67D.`
C. `-0.67D.`
D. `-37D.`

Explanation

To solve the problem of determining the power of the lens required for a nearsighted man who can clearly see objects only up to a distance of 1.5 meters, we can follow these steps:

### Step 1: Understand the Condition of Myopia
A nearsighted (myopic) person can see nearby objects clearly but struggles with distant objects. In this case, the maximum distance at which the man can see clearly is 1.5 meters.

### Step 2: Identify the Lens Type
To correct myopia, we need a diverging lens (concave lens) that will allow the person to see distant objects clearly. The lens will create a virtual image of distant objects (at infinity) at the distance where the person can see clearly (1.5 m).

### Step 3: Use the Lens Formula
The lens formula is given by:
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]
where:
- \( f \) is the focal length of the lens,
- \( v \) is the image distance (which will be -1.5 m for a virtual image),
- \( u \) is the object distance (which is considered to be at infinity for distant objects, so \( u = \infty \)).

### Step 4: Substitute the Values into the Lens Formula
Since the object distance \( u \) is at infinity, we have:
\[ \frac{1}{u} = 0 \]
Thus, the lens formula simplifies to:
\[ \frac{1}{f} = \frac{1}{v} - 0 \]
Substituting \( v = -1.5 \) m:
\[ \frac{1}{f} = \frac{1}{-1.5} \]
This gives:
\[ f = -1.5 \text{ m} \]

### Step 5: Calculate the Power of the Lens
The power \( P \) of a lens is given by the formula:
\[ P = \frac{1}{f} \text{ (in meters)} \]
Substituting \( f = -1.5 \) m:
\[ P = \frac{1}{-1.5} = -\frac{2}{3} \text{ diopters} \]
Calculating this gives:
\[ P = -0.67 \text{ diopters} \]

### Conclusion
The power of the lens required to correct the nearsightedness of the man is approximately **-0.67 diopters**.

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